The Spectral Schwartz Distribution

Wilhelm von Waldenfels

The spectral Schwartz distribution is connected with

the spectral measure, but it is in some way more gen-

eral and in some way more restricted than the spectral

measure. After the definition of the spectral Schwartz

distribution and some properties we present some ex-

amples. We calculate the resolvents of some operators

with the help of Krein’s formula and calculate then the

spectral Schwartz distribution. The method, how to

do this, has been developed by Gariy Efimov and my-

self in 1995 in a paper about radiation transfer. But the

method is so obvious, that it could be known before.

Assume we have a Banach space V and denote by L(V )

the space of all bounded linear operators from V to V

provided with the usual operator norm, and an open set

G ⊂ C and a function R(z) : G → L(V ) satisfying the

resolvent equation

R(z1)−R(z2) = (z2 − z1)R(z1)R(z2)..

The function R(z) is holomorphic in G. The subspace

D = R(z)V

is a subset independent of z ∈ G. If R(z0) is injective

for one z0 ∈ G, then R(z) is injective for all z ∈ G and

there exists a mapping a : D → V such that

(z − a)R(z)f = f for f ∈ VR(z)(z − a)f = f for f ∈ D

or

R(z) = (z − a)−1

and

aR(z) = −1 + zR(z) ; R(z)a = −1 + zR(z).

The operator a is closed and R(z) is the resolvent of a.

If G ⊂ C is open and f : G → C, f(z) = f(x + iy) has a

continuous derivative, set

∂f =df

dz=

1

2

(∂f

∂x− i

∂f

∂y

)∂f =

df

dz=

1

2

(∂f

∂x+ i

∂f

∂y

).

The function f is holomorphic if and only if ∂f = 0.

In an analogous way one defines these derivatives for

Schwartz distributions.

The function z 7→ 1/z is locally integrable and one ob-

tains

∂(1/z) = πδ(z).

Assume f to be defined and holomorphic for the ele-ments x+ iy ∈ G, y 6= 0, and that f(x± i0) exists, then

∂f(x+ iy) = (i/2)(f(x+ i0)− f(x− i0))δ(y)

Proposition Assume a function R(z) : G → L(V ), de-fined and obeying the resolvent equation almost every-where and a subspace V0 ⊂ V such that z 7→ (f |R(z)|g)is locally integrable for all f, g ∈ V0 then

z1, z2 7→ (f |R(z1)R(z2)|g)

is also locally integrable as well and for the Schwartzderivatives one has the formula

∂1∂2(f |R(z1)R(z2)|g) = πδ(z1 − z2)∂(f |R(z1)|g)

Definition Under the assumptions of the last proposi-

tion we call

M = (1/π)∂R

defined scalarly for f, g ∈ V0 by

(f |M(z)|g) = (1/π)∂(f |R(z)|g)

the spectral Schwartz distribution of R.

Under the assumptions of the last proposition and under

the additional assumption, that R(z) is injective, denote

again by a the operator defined by the resolvent. Then

we have

aM(z) = zM(z)

or more precisely

(f |aM(z)|g) = z(f |M(z)|g).

We consider again a function z ∈ G 7→ R(z) obeyingthe resolvent equation with values in a Hilbert spaceV and assume z, z ∈ G,=z 6= 0, R(z)∗ = R(z) and R(z)injective, and that D = R(z)V is dense in V . Then R(z)an be extended to the set of all z with =z 6= 0. If z 7→(f |R(z)|g), for f, g ∈ V0, is locally integrable, then thereexists a positive measure µ on R with values in L(V )defined by the sesquilinear forms f, g ∈ V0 7→ (f |µ(x)|g),such that

(f |M(x+ iy)|g) = (f |µ(x)|g)δ(y).

and

(f |µ(x)|g) =1

2πi(f |(R(x− i0)−R(x+ i0)|g)

Example 1 Suppose we have a matrix A with only

single eigenvalues and with the resolvent

R(z) =1

z −A=∑i

1

z − λipi

where pi are the eigenprojectors, so that pipj = piδij.

Then

M(z) = (1/π)∂R(z) =∑i

δ(z − λi)pi.

The last equation also holds if A is nilpotent, e.g., A2 =

0. Then one has to take

R(z) =1

z+A

Pz2,

where P denotes the principal value. Then

M(z) = δ(z)−A∂δ(z).

Example 2. Consider the multiplication operator Ω in

L2(R), given by (Ωf)(ω) = ωf(ω). The resolvent

RΩ(z) = (z −Ω)−1

is holomorphic outside the real line. The domain of Ω is

the space D = RΩ(z)L2, the space of all L2 functions f ,

such that Ωf is square integrable. Here we have defined

Ωf for all functions in a natural way. For f, g ∈ C1c

(f |RΩ(x± i0)|g) =∫

dωf(ω)g(ω)P/(x−ω)∓ iπf(x)g(x).

So so

(f |µ(x)|g) = f(x)g(x).

The the generalized eigen functions are

δx(ω) = δ(x− ω).

Using the formalism of bra and ket vectors we obtain

µ(x) = |δx)(δx|.

We have

RΩ(z) =∫

dx1

z − xµ(x) =

∫dx

1

z − x|δx)(δx|.

We have

Ω|δx) = x|δx).

The eigenvectors δx form a generalized orthonormal ba-

sis,i.e.

(δx|δy) = δ(x− y)∫

dx|δx)(δx| = 1.

The first equation ( orthonormality)can be checked di-rectly . The second equation (completeness) says, that∫

dxµ(x) = 1.

Example 3(Quantum probability:The pure number pro-cess) We consider the function R(z) which obeys theresolvent equation.

R(z) = RΩ(z) +1

1 + iπσ(z)RΩ(z)|E〉〈E|RΩ(z)

with σ(z) = signum = z and E is the constant functionE(ω) = 1 The expression

RΩ(z)|E〉 : RΩ(z)|E〉(ω) = 1/(z − ω)

is well defined in L2 and

〈E|RΩ(z) = RΩ(z)|E〉

In order to formulate the operator H corresponding to

R(z) we define the subspace L ⊂ L2

L = f = RΩ(z)(cE + f) : f ∈ L2

or more explicitely f ∈ L iff

f(ω) =1

z − ω(c+ f(ω))

Call L† the set of all semilinear functionals L → C. A

semilinear functional ϕ is additiv and ϕ(cf) = cϕ(f) for

f ∈ C. By the scalar product 〈g|f〉 =∫

dωg(ω)f(ω) we

associate to any f ∈ L2 a semilinear functional ϕ on L,

ϕ(ξ) = 〈ξ|f〉

As L is dense in L2 the functional determines f . So we

may imbed L2 into L† and

L ⊂ L2 ⊂ L†.

Define the functionals 〈E| ∈ L∗ and also |E〉 ∈ L† by

〈E|f〉 = limr→∞

∫ r−rf(ω)dω = −iπcσ(z) +

∫ 1

z − ωf(ω)dω

and

〈f |E〉 = 〈E|f〉.

Define the operator

Ω : L→ L†

〈g|Ωf〉 = limr→∞

∫ r−r

dωg(ω)ωf(ω)

Ωf = −c|E〉 − f + zf.

The domain of the selfadjoint operator H is

D = R(z)H =

RΩ(z)(|f〉+

(E|RΩ(z)|f〉1 + iσ(z)π

|E〉 : f ∈ L2⊂ L

The Hamiltonian H is the restriction of

H = Ω + |E〉〈E|

to that domain. With the methods used before we

calculate

M(x+ iy) = µ(x)δ(y)

µ(x) =1

2πi(R(x− i0)−R(x+ i0)) = |αx〉〈αx|

|αx〉 = (1 + π2)−1/2( Px−Ω

|E〉+ |δx〉)

. The αx form a generalized orthonormal basis in L2.

Example 4(Quantum probability: The Heisenberg equa-tion of the amplified oscillator) The underlying Hilbertspace is

H = C⊕ L2(R)

with the scalar product

〈(c, f)|(c′, g)〉 = cc′+∫

dxf(x)g(x).

We consider the function

R(z) =

(0 00 RΩ(z)

)+

(1

−RΩ(z)|E)

)1

z − iπσ(z)(1, (E|RΩ(z)) .

Define the operator

H : C⊕ L → C⊕ L†

H =

(0 (E|−|E) Ω

),

We have to distinguish between right and left domain

Dl resp Dr of the operator H corresponding to R(z).

Dl = HR(z) =

ξ ∈ C⊕ L : ξ = c(1, (E|R(z)) + (0, (f |)

Dr = R(z)H =

ξ ∈ C⊕ L : ξ = c

( 1

−R(z)|E)

)+( 0

R(z)f

)

with c ∈ C, f ∈ L2. The Hamiltonian H is the restriction

of H to Dl resp. Dr.

. The matrix H is not symmetric but it obeys to the

equation

JHJ = H+

with

J =

(−1 00 1

)

The resolvent R(z) is holomorphic outside the real line

and the two simple poles ±iπ. The spectral Schwartz

distribution M(z) = (1/π)∂R(z) has the form

M(x+ iy) = µ(x)δ(y) + p iπδ(z − iπ) + p− iπδ(z + iπ)

with

µ(x) =1

2πi(R(x− i0)−R(x+ i0)) = |αx〉〈βx|

|αx〉 = (x2 + π2)−1/2

( 1

− Px−Ω|E〉

)+ x

( 0

|δx〉

)〈βx| = (x2 + π2)−1/2

(− (1, 〈E|

Px−Ω

) + x(0, 〈δx|))

and

p±iπ = |α±iπ〉〈β±iπ| |α±iπ〉 =( 1

−1

±iπ −Ω|E〉

)

〈β±iπ| = (1, 〈E|1

±iπ −Ω)

It is easy to check the biorthonormality relations

〈αx|βy〉 = δ(x− y)

〈αx|β±iπ〉 = 0 (α±iπ|βx〉 = 0

〈α± iπ|β±iπ〉 = 1 α± iπ|β∓iπ〉 = 0

and completeness ∫dzM(z) = 1.

Example 5(Radiation transfer:grey atmosphere) De-

note

B = k ∈ R : |k| ≥ 1

and consider the operator

A = D − |η)(χ|

where D is the multiplication operator.

Df(k) = kf(k)

η(k) = sign(k)|k|−1/2

χ(k) = 12|k|−1/2

The operator can be defined by the resolvent

R(z) =1

z −D−

1

z −D|η)

PC(z)

(χ|1

z −Dwith

C(z) = 1 + (χ|1

z −D|η).

Near the origin

C(z) = −(χ|D−3|η)z2 + · · · = −13z

2 + · · · .

This necesssitates the introduction of P. The spectrumconsists of B and the origin, where we have a double

pole. For x ∈ B

C(x+i0) = 1+(χ|P

x−D−iπ(χ|δx)(δx|η) = CR(x)−iπCI(x).

We obtain

µ(x) = 1x ∈ B|αx)(βx|+ p0δ(x)− a0δ′(x).

with

|αx) = (C2R + π2C2

I )−1/2(CR|δx)−

Px−D

|η)(χ|δx)

)

(β| = (C2R + π2C2

I )−1/2(CR(δx| − (δx|η)(χ|

Px−D

)

p0 = (χ|D−3|η)−1(D−2|η)(χ|D−1 + (D−1|η)(χ|D−2

)

a0 = (χ|D−3|η)−1(D−1|η)(χ|D−1

)

Literature

G.V.Efimov, W.v.W.,R.Wehrse 1994

v.W.2013